Question

the spinner is divided into 8 equal sections. which two events have the same probability? spinner image options: o p(gray), p(green) o p(red), p(gray) o p(gray), p(purple) o p(yellow), p(gray)

Explanation:

Step1: Count sections per color

Yellow: 1, Blue:1, Gray:2, Green:2, Red:1, Purple:1.

Step2: Calculate probabilities

For gray: \( \frac{2}{8}=\frac{1}{4} \), green: \( \frac{2}{8}=\frac{1}{4} \), red: \( \frac{1}{8} \), purple: \( \frac{1}{8} \), yellow: \( \frac{1}{8} \).

Step3: Compare probabilities

Only \( P(\text{gray}) \) and \( P(\text{green}) \) have equal probabilities (\( \frac{1}{4} \) each).

Answer:

First, let's analyze the spinner divided into 8 equal sections. Let's count the number of each color:

• Yellow: 1 section
• Blue: 1 section
• Gray: 2 sections
• Green: 2 sections
• Red: 1 section
• Purple: 1 section

Now let's calculate the probability for each event in the options:

Option 1: \( P(\text{gray}) \), \( P(\text{green}) \)

• Number of gray sections: 2, so \( P(\text{gray})=\frac{2}{8}=\frac{1}{4} \)
• Number of green sections: 2, so \( P(\text{green})=\frac{2}{8}=\frac{1}{4} \)

So \( P(\text{gray}) = P(\text{green}) \)

Option 2: \( P(\text{red}) \), \( P(\text{gray}) \)

• \( P(\text{red})=\frac{1}{8} \)
• \( P(\text{gray})=\frac{2}{8}=\frac{1}{4} \)

Not equal.

Option 3: \( P(\text{gray}) \), \( P(\text{purple}) \)

• \( P(\text{gray})=\frac{2}{8}=\frac{1}{4} \)
• \( P(\text{purple})=\frac{1}{8} \)

Not equal.

Option 4: \( P(\text{yellow}) \), \( P(\text{gray}) \)

• \( P(\text{yellow})=\frac{1}{8} \)
• \( P(\text{gray})=\frac{2}{8}=\frac{1}{4} \)

Not equal.

So the two events with the same probability are \( P(\text{gray}) \) and \( P(\text{green}) \), which corresponds to the first option (the option with \( P(\text{gray}), P(\text{green}) \)). If we assume the first option is the one labeled with \( P(\text{gray}), P(\text{green}) \) (the first circular option), then the answer is that option. If we need to write the option as per the given choices (assuming the first option is the one with \( P(\text{gray}), P(\text{green}) \)):

The correct option is the first one (e.g., if the options are labeled as A, B, C, D with A being \( P(\text{gray}), P(\text{green}) \), then A. \( P(\text{gray}), P(\text{green}) \))